If `p-1/(p)=4`, find the value of `p^(4)+1/(p^(4))`.

Answer

To solve the problem, we need to find the value of \( p^4 + \frac{1}{p^4} \) given that \( p - \frac{1}{p} = 4 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ p - \frac{1}{p} = 4 \]

2. **Square both sides:**
\[ \left(p - \frac{1}{p}\right)^2 = 4^2 \]
This simplifies to:
\[ p^2 - 2\left(p \cdot \frac{1}{p}\right) + \frac{1}{p^2} = 16 \]
Which can be rewritten as:
\[ p^2 - 2 + \frac{1}{p^2} = 16 \]

3. **Rearranging the equation:**
\[ p^2 + \frac{1}{p^2} - 2 = 16 \]
Adding 2 to both sides gives:
\[ p^2 + \frac{1}{p^2} = 16 + 2 = 18 \]

4. **Now, we need to find \( p^4 + \frac{1}{p^4} \). We can use the identity:**
\[ \left(p^2 + \frac{1}{p^2}\right)^2 = p^4 + 2 + \frac{1}{p^4} \]
Substituting \( p^2 + \frac{1}{p^2} = 18 \):
\[ 18^2 = p^4 + 2 + \frac{1}{p^4} \]
This simplifies to:
\[ 324 = p^4 + 2 + \frac{1}{p^4} \]

5. **Rearranging to find \( p^4 + \frac{1}{p^4} \):**
\[ p^4 + \frac{1}{p^4} = 324 - 2 = 322 \]

### Final Answer:
\[ p^4 + \frac{1}{p^4} = 322 \]